Exact regular polygon-count A-infinity branch

Write ω=dλ and ${\iota }_{X_H}\omega =-dH$, as in the actual generator Definition. A polygon has one negative/output strip end and k positive/input strip ends, boundary labels L0,…,Lk, and the actual prescribed transverse intersection or nondegenerate chord generators. Data on the universal punctured-disk family consist of $J_z$, tamed by ω, and K∈Ω¹(domain;C∞(X)). On each boundary arc K vanishes on its tangent, and on an end it is $H_t$ dt with $J_z=J_t$. The equation is $(du-X_K{)}^{0,1}=0$. In the intersection model take $H=0$. “Admissible” here spells out the data already required to define moduli up to domain automorphism: data are equivariant under those automorphisms, coherent over the compactified real disk-domain family, and equal to the product end data on gluing collars. Thus for a two-punctured strip they are translation invariant; a continuation equation depending arbitrarily on s is not a differential polygon datum. For a zero-input disk, its affine translation/dilation automorphisms force $K=0$: in an upper-half-plane coordinate invariance makes its two coefficients depend only on height and scale as ${\mathrm{height}}^{-1}$, and smoothness at ordinary boundary points forces both coefficients to vanish. Its J is constant under this transitive automorphism action. Thus a zero-input bubble is an ordinary holomorphic disk, rather than an arbitrarily assigned continuation equation. No escape is the given uniform compact image condition, not an unproved maximum principle.

Likewise the existing “transverse morphism generators” hypothesis is the full relevant asymptotic generator system of the declared model, not merely the external two endpoints with unrecorded degenerate intermediate chords. Only path components that occur in this polygon problem are relevant. The current target's Given already defines regularity as surjectivity of EVERY relevant linearized operator. In applying it below, this includes component problems appearing in its zero/one-dimensional compactification. A negative-index regular component is therefore absent. Coherent orientations mean the given determinant orientations compatible with these actual gluing maps; they are not deduced merely from the word “relatively spin.” These distinctions are essential: the earlier infinite-strip-breaking counterexample applies to the weaker endpoints-only interpretation.

The complete linear suppliers are the attached [F1] and [F2]. Finite-chart core parametrices and their actual onto weighted end inverse give Fredholm operators on punctured bordered domains. Compact-core parametrix errors gain a derivative and are compact; end coefficient errors decay and are small on tails. This gives a two-sided parametrix, smooth kernels and cokernels, and weighted regularity. No selfadjoint asymptotic operator or implicit gluing estimate is being assumed.

Put $v_s=u_s-X_{K_s}$, $v_t=u_t-X_{K_t}$. The equation is $v_s+J_z{v}_t=0$. With the symmetrized tame Hermitian metric, its geometric energy is $\int \omega (v_s,v_t)\ge 0$. Direct substitution using ${\iota }_{X_H}\omega =-dH$ gives $\omega (v_s,v_t)=u\ast \omega (s,t)-d(u\ast K)(s,t)+R_K(s,t,u),$ $R_K=\mathrm{\partial }s{K}_t-\mathrm{\partial }t{K}_s+\omega (X_{K_s},X_{K_t})\mathrm{.}$ Indeed the two mixed terms are $d_X{K}_s(u_t)-d_X{K}_t(u_s)$, which are exactly the difference between the exterior derivative along u and the explicit source derivative. Integrate on a truncated punctured domain and apply Stokes to λ−K. With $A_H(\gamma )={\int }_0^1[\lambda (\gamma dot)-H_t(\gamma )]dt+f_{s}tart(\gamma (0))-f_{e}nd(\gamma (1)),$ the exact boundary primitives cancel along every $L_i$ arc, leaving $E(u)={\sum }_{i}nputsA({\gamma }_i)-A({\gamma }_0)+\int R_K(z,u(z))\mathrm{.}$ This fixes the signs: on a strip the positive end contributes its action and the negative end its negative action. The curvature is zero on all product necks/ends. Coherent smooth data on the compact disk-domain family have uniformly bounded curvature integrals on the remaining cores, for images in the GIVEN compact no-escape set. Thus energy is uniformly bounded for each ordered generator word. This is the missing actual inhomogeneous energy estimate.

First justify the boundary estimates for the actual source-dependent data; freezing after rescaling alone would not give the prelimit bounds. In a finite smooth core-family chart write its parameters as $a\in {\mathbb{R}}^d$ and its source coordinate as $z=s+it$, with boundary $t=0$. Choose smaller convex parameter/source charts with compact closure inside larger charts. On the complex torus $Q=(\mathbb{C}\mathrm{/}(\mathbb{Z}+i\mathbb{Z}){)}^{d+1}$ choose a smooth map into the larger chart equal to $(a,z)$ on the smaller one: multiply local coordinates minus the chart center by a cutoff and add the center, using convexity. Pull $J_{a,z}(x)$ back by this map. Extend the Hamiltonian-vector-field-valued one-form $W=X_K$ from the smaller chart to $Q$ by a cutoff, with zero auxiliary parameter components there. For $\eta \in TQ$, set $C\eta =W(i_Q\eta )-JW(\eta )$ and $\mathbb{J}(\eta ,v)=(i_Q\eta ,Jv+C\eta )$. Since $C{i}_Q+JC=0$, this is one fixed smooth almost-complex structure on $Q\times X$. The graph $U(z)=(a,z,u(z))$, with constant $a$, is $\mathbb{J}$-holomorphic exactly when $(du-X_K{)}^{0,1}=0$. Its boundary lies on the compact ${\widehat{L}}_i={\mathrm{\Gamma }}^{d+1}\times L_i$, where $\mathrm{\Gamma }=\mathbb{R}\mathrm{/}\mathbb{Z}$ is the real torus circle. This is totally real: tangency of a vector and its $\mathbb{J}$ image forces its horizontal component to lie in $T{\mathrm{\Gamma }}^{d+1}\cap i_{Q}T{\mathrm{\Gamma }}^{d+1}=0$, then its vertical component lies in $T{L}_i\cap JT{L}_i=0$. The restriction of this autonomous target structure to ${\widehat{L}}_i$ is therefore a fixed boundary structure, including when the original parameter $a$ varies between maps. The local proofs in [F5] now apply to these graphs. Their compactness uses are precisely a finite adapted-metric cover of the compact boundary submanifold, bounded coefficient derivatives on a relatively compact neighborhood of the compact image, and subsequence extraction in that compact image; all hold on $Q\times X$ by the given no-escape set, without asserting compactness of $X$. Use a fixed product target metric. Its graph energy is $E_g(u)$ plus the flat source area, bounded by 4.1 and the uniformly bounded integral of $\mathrm{\mid }{X}_K{\mathrm{\mid }}^2$; the extra area on a radius-$r$ patch is $O(r^2)$. The normalized extraction, bootstrap and removal arguments of [F5] consequently give a nonconstant finite-energy sphere or disk bubble at any gradient blowup. Its horizontal component is constant: the $a$ component is already constant and the source-coordinate derivative is multiplied by the shrinking rescaling factor. It is thus a vertical bubble for the frozen $J_{a_{\ast },z_{\ast }}$, with disk boundary on the single $L_i$. Exactness excludes it: Stokes gives sphere area zero or disk area ${\int }_{\mathrm{\partial }D}d{f}_i=0$, whereas tameness gives strictly positive area for a nonconstant holomorphic map. Thus compact-core gradients are bounded. The graph bootstrap and projection to $X$ give smooth convergence on every compact core after the finite domain/translation extractions. This construction verifies the boundary hypothesis before applying the estimates and incorporates the forcing in the fixed graph structure.

The attached [F3] constructs the compact boundary-pointed disk-domain space DIRECTLY: normalize three real anchors, repeatedly rescale coinciding finite groups, and recover the unique finite stable planar tree by its finitely many cross-ratios. Its local family has the actual real node fixture zw=−ρ, ρ≥0, with strip length−π^−1logρ, and proper compact cores. It supplies the finite associahedral product strata without assuming general algebraic/analytic curve compactification. Constant unmarked sphere trees contract, and there are no positive-energy sphere components by 5.1. Two-punctured nonconstant strip components are recorded separately by their translation gauges; trivial constant-chord strips are unstable and are not counted.

The given full relevant transverse generator set is finite on the compact image set. For intersections this follows from compactness and transversality. For chords, any convergent sequence with images in that compact set has a limiting chord by the ODE equation, and nondegeneracy makes the relevant set discrete. On each end conjugate by its Hamiltonian flow. The equation then becomes an actual almost-complex strip with transverse Lagrangian edges, and the Floer energy is uniformly comparable to its ordinary energy near the chord. The local action/neck estimates and the already proved finite transverse-strip-breaking argument apply on this compact geometry. Nonconstant strip energies have a common positive lower bound: on an exact autonomous end they are positive differences of the FINITELY many chord actions. A zero difference forces zero energy and a trivial strip. Thus only finitely many nonconstant strips can be extracted.

More explicitly, select disjoint fixed-width end windows with energy at least a small epsilon threshold. There are at most E0/epsilon of them. Cluster centers at bounded mutual distances and translate one representative per cluster. Compact-core smooth extraction gives actual finite-energy Floer strips. Between clusters every fixed-radius patch has small energy; the actual transverse-strip local-action lemma gives exponential energy and image collapse, uniformly in the remaining length. The tame version uses the same actual local calculation: in the simultaneous transverse chart choose a primitive vanishing on both Lagrangian axes; its action satisfies $a^{\mathrm{\prime }}(s)=-\int \omega (v_s,v_t)$ and $\mathrm{\mid }a(s)\mathrm{\mid }\le C\int \mathrm{\mid }{v}_t{\mathrm{\mid }}^2$. Tameness compares the latter with slice energy. For central neck energy e(R), Stokes then gives e(R)≤−C e'(R), hence exponential decay; local epsilon and bootstrap give all higher derivative decay and image collapse. Compatibility is not used in this inequality. In Hamiltonian coordinates the collapse is to the constant intersection; before conjugation it is to its chord. Thus adjacent ends match and no energy is lost. This also treats each long domain-degeneration strip from 6.1. Exactness excludes all remaining rescaled bubbles by 5.1. A finite partition into core pieces, strip clusters and the collapsing intervening regions gives the full energy identity, before passing to the finite cutoff limits. Boundary labels and original corners remain distinct by the real stable-domain construction. Additional constant unstable pieces contract with their inherited matching values. This proves the required polygon compactness, rather than invoking a no-escape assumption as if it already were compactness.

Work first at two regular component solutions with one matching nondegenerate generator. Fix the three domain normalization points on a stable component. On a nonconstant two-punctured strip fix its translation by one scalar core evaluation transverse to $u_s$, which is nonzero somewhere; the ordinary inverse theorem gives the local gauge. Subtracting this infinitesimal kernel direction corrects that gauge without losing operator surjectivity. Let $D_i:X_i\to Y_i$ be the ACTUAL full component derivative, including its finite domain parameters, in small exponential-weight Hilbert spaces, with kernel $K_i$ after these gauges. Regularity is precisely its assumed onto property. Choose finite core/E/domain evaluations $L_i$ identifying $K_i$, and let $Q_i$ be its inverse on kerL_i. All these assertions follow from the actual Fredholm suppliers in 3.1 and the bounded inverse theorem. The kernel sections have a uniform exponential rate ν larger than the gluing weight: cut off on a sufficiently far end and write its equation as $(P_0+B)\xi =\mathrm{core}$ forcing with B small H1→L2. The explicit end inverse is available both unweighted and with weight exp(ν $\max (s-s_0,0)$); Neumann inversion gives a weighted solution, which is also unweighted because the weight is at least one. Unweighted uniqueness identifies it with the original cutoff section. Higher strip estimates give the same rate in every required fixed derivative order. Thus this rate is obtained from the constructed end inverse rather than assumed as a spectral theorem.

The component inverse constants can be chosen uniformly as the positive weight tends to zero. The transverse end inverses for all weights in a small interval about zero give the same compact-core Fredholm realization: applying the cutoff argument of step 9.1 also to the adjoint shows that both kernel sections and adjoint annihilators decay at a rate larger than these weights. Their finite kernel and obstruction spaces are therefore unchanged, so the given surjectivity holds also at weight zero. At weight zero, augment this onto Fredholm component operator by the fixed core coordinates $L_i$; $(D_i,L_i)$ is an isomorphism. Identify a weight-$\delta$ domain/target with the unweighted spaces by multiplication by $e^{\delta \beta }$, where $\beta$ is zero on the core and equals end distance on each end, with bounded derivatives. Conjugation changes the first-order operator by a multiplication operator of norm at most $C\mathrm{\mid }\delta \mathrm{\mid }$ from $H^k$ to $H^{k-1}$; core/domain-parameter terms and $L_i$ are unchanged. The unweighted inverse therefore gives a Neumann inverse for all $\mathrm{\mid }\delta \mathrm{\mid }\le {\delta }_{\ast }$, after decreasing ${\delta }_{\ast }>0$, with one common bound. Restricting the augmented inverse to zero $L_i$ coordinate gives the required uniformly bounded $Q_i$. The end inverse in [F1] and its small-weight perturbation give the same uniform bounds in end estimates. Consequently the constants multiplying $W^{-1}{e}^{2\delta W}$ below are bounded independently of sufficiently small $\delta$.

Near the common chord use Hamiltonian coordinates and a smooth simultaneous chart straightening the two transverse Lagrangian submanifolds into their linear coordinate axes. Such a chart is obtained from their two transverse local parametrizations and the ordinary inverse theorem; it need not be symplectic. The nonlinear boundary condition is now linear on each strip edge. Glue the two ends at length2R, retain each component to depth R/2, and cut its exponentially decaying remainder to zero before the middle. Boundary values stay on their axes. The limiting chord is fixed in the middle. The operator coefficients converge exponentially to the explicit transverse end operator, and its weighted inverse was CONSTRUCTED in the strip supplier. For some ν>0 and weight0<δ<ν, direct differentiation gives $\mathrm{\parallel }{F}_R(0)\mathrm{\parallel }+\mathrm{\parallel }{D}_R({\kappa }_R)\mathrm{\parallel }\mathrm{/}\mathrm{\parallel }\kappa \mathrm{\parallel }\le C{e}^{-(\nu -\delta )R\mathrm{/}2}\mathrm{.}$ Here κ_R is the preglued kernel direction and $L_R{\kappa }_R=(L_1⨁L_2)\kappa$ exactly, because all L-coordinates are on unchanged cores. Product perturbation data on the glued neck make its curvature/error zero there; the only residuals are the displayed cutoff and exponentially small coefficient errors.

Build a bounded approximate right inverse, not an assumed one. Break a target forcing at the middle R with complementary fixed-width cutoffs, extend each part by zero on its infinite component end, and apply $Q_1,Q_2$. Glue their section remainders by cutting the left one only near3R/2 and the right one only nearR/2, retaining both in the middle. There is no shared constant section mode: transversality makes the end intersection zero. The nodal end weight at3R/2 is $e^{3\delta R\mathrm{/}2}$, while the glued weight there is $e^{\delta R\mathrm{/}2}$, giving cutoff error $C{e}^{-\delta R}$; the other side is identical. Core L-coordinates remain zero. Coefficient-comparison errors are $C{e}^{-\nu R\mathrm{/}2}$. Thus $\mathrm{\parallel }{T}_R\mathrm{\parallel }\le C,\mathrm{\parallel }{D}_R{T}_R-I\mathrm{\parallel }\le C[e^{-\delta R}+e^{-\nu R\mathrm{/}2}]\mathrm{.}$ Neumann inversion gives an actual uniformly bounded $Q_R$ with image in kerL_R.

Prove this complement is the FULL complement; do not assume an index-gluing theorem. Break a section in kerL_R with cutoffs varying over a fixed width W around R. Its broken norms and the original norm are comparable by $e^{2\delta W}$. Component inversion by $Q_i$ bounds the broken sections by the broken forcing plus C $W^{-1}{e}^{2\delta W}$ times the original section norm and the exponentially small coefficient error. Use the uniform inverse bound of step 10.1: choose W large enough that C $e\mathrm{/}W<1\mathrm{/}4$, then choose 0<δ≤δ_* with 2δW≤1 and δ<ν, and finally choose R large. Absorb the error. This proves $\mathrm{\parallel }\xi \mathrm{\parallel }\le C\mathrm{\parallel }{D}_R\xi \mathrm{\parallel }onker{L}_R\mathrm{.}$ The right inverse in 11.1 therefore has image exactly kerL_R. For any $\kappa \in K_1⨁K_2$, ${\kappa }_R-Q_R{D}_R{\kappa }_R$ lies in kerD_R and has the prescribed L-coordinate. The estimate gives uniqueness. Hence kerD_R is exactly this copy of $K_1⨁K_2$. This proves the required linear gluing dimension and uniform splitting from actual cut estimates.

Choose a source-dependent retraction/exponential chart adapted to each boundary $L_i$ on its disjoint source arc neighborhoods; on the neck use the straight axes chart. This avoids assuming intersecting different Lagrangians are simultaneously totally geodesic for one metric. Fixed-radius half-strip Sobolev estimates and real reflections give the uniform $H^k$ product estimates for k≥4. All coefficient derivatives on the compact cores are bounded, and every neck has the same bounded unit-patch geometry. Taylor's integral remainder therefore gives $\mathrm{\parallel }{F}_R(\xi )-F_R(\eta )-D_R(\xi -\eta )\mathrm{\parallel }\le C(\mathrm{\parallel }\xi \mathrm{\parallel }+\mathrm{\parallel }\eta \mathrm{\parallel })\mathrm{\parallel }\xi -\eta \mathrm{\parallel },$ with C independent of R. The actual residual in 10.1 and right inverse in 11.1 now give a contraction on κ_R+imQ_R, with Lipschitz constant below1/2 in one fixed small ball. It has one and only one solution, whose correction norm is at most $2\mathrm{\mid }\mathrm{\mid }{Q}_R\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }{F}_R({\kappa }_R)\mathrm{\mid }\mathrm{\mid }$. This constructs the glued family for large R, with its exact kernel coordinates.

Conversely, a sequence converging to this broken pair has smooth core convergence by 5.1 and exponential end collapse by 8.1. In the same axes coordinates its difference from the preglued map tends to zero in the weighted norm: choose a fixed large end cutoff first, use core convergence there, and integrate the uniform decay against a weight smaller than ν on the remaining neck; then let that cutoff tend to infinity. Its L-coordinates determine κ. The uniqueness in 13.1 identifies it with the glued family. The same contraction comparison proves continuity as R→∞ and for finite component parameters. Thus $(\rho =e^{-R},\kappa )$ is an actual local collar chart, not only a formal smoothing. It is not claimed differentiable in ρ at zero. For finitely many breaks use disjoint end cutoffs and the direct sum of the component $Q_i$ and $L_i$. The error bounds add over these disjoint ends and tend to zero with the minimum neck length; the cutoff coercivity and contraction are unchanged. This gives the full product of r half-collar parameters and the component kernel coordinates, including the partial-breaking strata. Its positive-length image is open by the same inverse-gluing argument. Thus any nearby solution has actual local dimension Σ dimK_i+r. All operations are local in the compact no-escape region, so the noncompact ambient target introduces no extra step.

The compact bordered Maslov index, with the transverse chord capping operators, gives the cohomological polygon dimension $d(x0;x1,\dots ,xk)=\mathrm{\mid }x0\mathrm{\mid }-\sum \mathrm{\mid }xi\mathrm{\mid }+k-2.$ Here |x| is the UNshifted cohomological Floer degree. One exact normalization is i(P0#,P1#)=n+θ1−θ0−2Σα_j for graded transverse planes, where 0<α_j<1/2 are their unitary positive angles and exp(2πiθ_j) their squared phases. The capping normalization is computed on the unitary diagonal scalar planes: Fourier modes on the interval have real exponents θ+πm, so a crossing of one angular multiple of π changes the cap index by one. Between crossings the explicit weighted end inverse keeps the index constant. The integer n+θ1−θ0−2Σα_j records exactly these crossings and the n positive real reference directions. Closing the corner paths with these cappings gives the doubled Maslov degree; its index is the bordered index in 3.1. Additivity when attaching caps follows from the same cut/right-inverse construction in steps 9.1–14.1: first add finitely many supported forcing directions to make each cap operator onto, perform that proved construction, and subtract the added dimensions. Inputs and output cappings occur with opposite orientations, yielding the displayed difference. This is also the determinant-line grading convention for the generators. The k−2 term is the domain dimension; for a two-punctured strip it is −1, its translation quotient. For a tree with r breaks, cancellation of the internal generator degrees gives $d_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}={\sum }_v{d}_v+r$, also directly recovered by the kernel/length construction in 12.1.

Every component operator relevant to the compactification is regular by the given hypothesis in 2.1, so its actual dimension equals its nonnegative index. The multi-collar in 14.1 gives an OPEN positive-length solution neighborhood of dimension Σ dimK_i+r. A point from the original zero/one-dimensional regular moduli must have that same local dimension, by uniqueness of manifold dimension (or its local homology). Hence a zero-dimensional moduli space has no break; by steps 4.1–8.1 it is compact and discrete, hence finite. In a one-dimensional space, a broken configuration has exactly one break and both factors are zero dimensional. Multiple breaks would require a negative-dimensional regular component. The collar theorem in steps 9.1–14.1 makes each such point a genuine boundary point; steps 4.1–8.1 show there are no other ends. Thus its compactification is a compact oriented topological one-manifold with precisely the asserted broken pairs. Coherent orientations are used here as GIVEN, not proved from relative spin.

We record the domain sign rather than hiding it under a gluing slogan. Normalize output∞, first input0 and last input1; orient the stable k-input domain by dx2∧…∧dx_(k−1). When a consecutive block of s inputs shrinks after r inputs and before t inputs, $r+s+t=k$, write its points as $a+\rho b_j$ with b1=0,bs=1. Direct wedge expansion gives ${\mathrm{\Omega }}_k=(-1{)}^{r+st}{\rho }^{s-2}d\rho \wedge {\mathrm{\Omega }}_{o}uter\wedge {\mathrm{\Omega }}_{i}nner\mathrm{.}$ For an interior block, move dρ past its r preceding coordinates and then move the t−1 after-coordinates past the s−2 inner coordinates; the exponent is r+t(s−2), equal mod2 to r+st. If $r=0$ use $\rho =x_s$; if $t=0$ use $\rho =1-x_{r+1}$; the same expansion gives exponents st and r respectively. Outward normal is−∂ρ, producing the additional common minus sign in the boundary identity. For strip ends the same normal corresponds to +∂R, and the given coherent determinant gluing with the chosen positive translation gauge supplies its identical graded contraction rule.

Let s denote the degree−1 suspension and set $b_k=(-1{)}^{k(k-1)\mathrm{/}2}s{\mu }_k(s^{-1}{)}^{\otimes k}$. Its coefficient sign is ${\beta }_k={\sum }_j(k-j)\mathrm{\mid }{x}_j\mathrm{\mid }$: the tensor Koszul sign contributes ${\sum }_j(k-j)(\mathrm{\mid }{x}_j\mathrm{\mid }-1)$ and the explicit arity factor supplies the difference. Moving the inner degree-one $b_s$ past r preceding shifted orientation lines gives precisely $(-1{)}^{{\sum }_{j\le r}(\mathrm{\mid }{x}_j\mathrm{\mid }-1)}$; the matching intermediate generator line and its dual cancel. Explicitly, with $Q={\sum }_{j\le r}\mathrm{\mid }{x}_j\mathrm{\mid }$, the inner degree $2-s$ contributes $(2-s)Q$ before suspension and ${\beta }_s+{\beta }_{r+1+t}-{\beta }_k\equiv st+(1-s)Q$. Adding the domain/outward sign $r+st+1$ leaves $1+{\sum }_{j\le r}(\mathrm{\mid }{x}_j\mathrm{\mid }-1)$. Removing its one common outward minus in the boundary equation gives the claimed shifted insertion sign. Hence the signed boundary count of 16.1 is exactly ${\sum }_{r+s+t=k}(-1{)}^{{\sum }_{j\le r}(\mathrm{\mid }{x}_j\mathrm{\mid }-1)}$ $b_{r+1+t}(x1,\dots ,xr,b_s(x_{r+1},\dots ,x_{r+s}),\dots ,xk)=0.$ The arity-one terms are the input/output strip breaks. A zero-input holomorphic disk bubble is constant by exactness and unstable, so no such extra face survives in this exact polygon regime. If curvature is recorded, it is zero for these genuine exact disk counts. Over F2 all signs are suppressed. In characteristic different from2 the displayed convention fixes them; no claim of orientation existence beyond the given coherent system is made.

The rigid counts are finite for each generator word and therefore define the unweighted operations. For a weighted version use the actual symplectic area $A_{\omega }(u)=\int u^{\ast }\omega$, not the geometric energy $E(u)$ of step 4.1. Stokes on the truncated domain, followed by exponential end convergence, gives $A_{\omega }(u)={\sum }_{i=1}^{k}G({\gamma }_i)-G({\gamma }_0)$, where $G(\gamma )={\int }_0^1\lambda (\dot{\gamma }(t))dt+f_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}(\gamma (0))-f_{\mathrm{e}\mathrm{n}\mathrm{d}}(\gamma (1))$; at an intersection take the constant chord. Thus $A_{\omega }$ is constant on the entire moduli space with that generator word. At a broken pair the two occurrences of $G$ of the matching generator cancel, so its two area exponents add to this same constant. Consequently weighting each rigid count by $T^{A_{\omega }(u)}$ multiplies every term of its fixed-word A-infinity relation by the same monomial and proves the identity coefficientwise. In the energy-only coefficient convention of [F4], take the additive exponent group $\mathrm{\Gamma }=\mathbb{R}$ with valuation $\nu (a)=a$, and write $T^a$ for its monomial: the full completion consists of sums ${\sum }_a{c}_a{T}^a$ with finitely many nonzero terms below each real cutoff. Its nonnegative subring permits these area weights only when all contributing areas are nonnegative; the full completion also permits negative exponents. Each operation has a finite generator basis and finitely many coefficients for its fixed arity, so these formulas extend continuously to the corresponding completed inputs. In the unperturbed intersection model $A_{\omega }=E\ge 0$. For general Hamiltonian data geometric energy instead equals $A_{\omega }(u)-{\sum }_i{\int }_0^1{H}_i(t,{\gamma }_i(t))dt+{\int }_0^1{H}_0(t,{\gamma }_0(t))dt+\int R_K(z,u(z))$, so it need not be locally constant and is not used as a coefficient exponent. No weighted assertion for $T^{E(u)}$ follows without an additional constancy hypothesis. The unweighted identity in all the stated regimes follows from the total boundary of a compact oriented one-manifold, whose intervals have opposite endpoint signs and whose circles have none. This proves the target boundary formula and A-infinity identities under the original declared hypotheses, without assuming the compactification or collars as additional inputs. It proves neither existence of regular perturbation data, a relative-spin orientation theorem, nor the general FOOO virtual-chain construction.